Characterizes alternating links in thickened surfaces using Gordon-Litherland pairing.
problem Identifying alternating links in thickened surfaces.
method Extension of Gordon-Litherland pairing to thickened surfaces.
result A non-split link in a thickened surface is alternating if and only if it bounds two definite surfaces of opposite sign.
Study minimal surfaces in Kropina 3D space, finding only planes as minimal translation surfaces.
problem Characterizing minimal surfaces in Kropina 3D space.
method Solving partial differential equations to characterize minimal surfaces.
result Only planes are minimal translation surfaces in Kropina 3D space.
Characterizes Bonnet surfaces using analytic conditions.
problem Characterizing Bonnet surfaces by geometric conditions.
method Derives Bonnet surfaces using two analytic conditions: mean curvature reduction to an ODE and the Painlevé property.
result Bonnet surfaces characterized by analytic conditions.
Characterizes quasiconformally homogeneous ladder surfaces.
problem Characterize quasiconformally homogeneous Riemann surfaces.
method Breaks the problem into four cases and proves one case.
result Every quasiconformally homogeneous ladder surface is quasiconformally equivalent to a regular cover of a closed surface.
In this study, we give the dual characterizations of Mannheim offsets of the ruled surface in terms of their integral invariants and the new characterization of the Mannheim offsets of developable surface. Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets …
Characterizes concircular helices and surfaces in 3D space.
problem Understanding concircular helices and surfaces in Euclidean 3-space.
method Characterization through differential equations and ruled surfaces.
result Characterizes concircular helices and surfaces in 3D space.
The study characterizes minimal surfaces in a specific three-dimensional metric space and finds that planes are the only minimal surfaces.
problem Characterizing minimal surfaces in a specific three-dimensional metric space.
method Obtained partial differential equations to characterize minimal surfaces and proved that planes are the only such surfaces.
result Planes are the only minimal surfaces in the Matsumoto space.
Characterizes knot groups and symmetric quandles of surface-links.
problem Characterize knot groups and symmetric quandles of surface-links.
method Used plat presentations for surface-links and closed 2-dimensional braids.
result Generalized results to include non-orientable surface-links and showed that dihedral quandles can be realized as symmetric quandles of surface-links.
Defines surfaces with specific contour patterns.
problem Characterizing surfaces with concentric or parallel K-contours. method Presented examples and characterization given.
result Characterization of surfaces with concentric or parallel K-contours. Characterizes polygonal surfaces in pseudo-hyperbolic spaces.
problem Understanding polygonal surfaces in pseudo-hyperbolic spaces.
method Characterizes polygonal surfaces by total curvature finiteness and asymptotic flatness, using comparison of ideal boundaries.
result Polygonal surfaces have parabolic type and polynomial quartic differential.
Characterizes multitwists in surface mapping class groups.
problem Understanding multitwists in surface mapping class groups.
method Characterization through braid relations.
result Characterized multitwists satisfying braid relations.
New method characterizes surface quadrilateral layouts as special immersions.
problem Characterize surface quadrilateral layouts mathematically.
method Characterizes quadrilateral layouts as special immersions of a cut representation of the surface into the Euclidean plane.
result Mathematically describes and generalizes integer grid maps.
For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are charac…
Characterizes special curves on surface tangent bundles.
problem Understanding curves on surface tangent bundles.
method Characterization of Legendre and slant curves.
result Characterizations for N-Legendre and N-slant curves.
We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in R2,1 to other Lorentzian space forms. We also characterize immersions of Riemannia…
Characterizes stable minimal capillary surfaces with specific angles.
problem Understanding stable minimal capillary surfaces with near 0 or π angles. method Curvature estimates for sequences of weakly stable minimal capillary surfaces.
result Characterization of tangential limits of stable minimal capillary surfaces.
Random walk constructs Morse functions on surfaces.
problem Creating Morse functions on surfaces.
method Random walk method to construct Morse functions.
result Small set of Morse functions approximates any other function.
Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address developable surfaces and ruled surfaces. The characterizations are higher order nonlinear PDEs generalizing the ones by Gaus…
A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…
We characterize geometrically the Lyapunov exponents of a cocycle (of arbitrary rank) with respect to a harmonic current defined on a hyperbolic Riemann surface lamination. Our characterizations are formulated in terms of the expansion rates of the cocycle along geodesic rays.
In this study, we give the relationships between the conical curvatures of ruled surfaces drawn by the unit vectors of the ruling, central normal and central tangent of a regular ruled surface in the Euclidean -space. We obtain the differential equations characterizing slant ruled surfaces and if the reference ruled su…
The paper characterizes surfaces in 4D space forms with flat normal connection.
problem Characterizing surfaces in 4D space forms with specific geometric properties.
method Analyzing linearly dependent conditions and using properties of sectional curvature.
result Characterizations of space-like and time-like surfaces with flat normal connection.
Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory bett…
Characterizes parabolic flute surfaces with specific parameters.
problem Characterizing flute surfaces with ergodic geodesic flow.
method Extending results on Fenchel-Nielsen coordinates and analyzing twist parameters.
result Characterizes parabolic flute surfaces with twist parameters in {0,1/2}.
Characterizes curves for minimal surfaces in de Sitter space.
problem Minimal surfaces in de Sitter space.
method Variational problem to find critical points of center of mass.
result Curves are critical points of center of mass.
New minimal surfaces desingularize three Clifford tori, proving uniqueness and characterizing them.
problem Desingularizing the union of three Clifford tori in a three-sphere.
method Constructing minimal surfaces with hexagonal boundary under group action, proving uniqueness.
result Characterized and proved uniqueness of desingularized surfaces.
Weakly Einstein Kähler surfaces are characterized and classified.
problem Characterizing and classifying weakly Einstein Kähler surfaces.
method Several conditions and constructions to characterize and classify weakly Einstein Kähler surfaces.
result Classification of weakly Einstein Kähler surfaces with specific properties and construction of new examples.
In a three-dimensional Riemannian manifold M that admits a unit Killing vector field ξ, we regard ξ as a magnetic vector field. A magnetic Hopf surface is a surface obtained by Lie dragging the magnetic curve with ξ. Then we characterize Sasakian structure on M from magnetic Hopf surfaces. That is, we show that i…
The paper characterizes biconservative surfaces in hyperbolic 4-space.
problem Characterizing biconservative surfaces in hyperbolic 4-space.
method Establishing intrinsic conditions and analyzing geometric properties.
result Biconservative surfaces in hyperbolic 4-space satisfy a specific intrinsic condition.
For non-degenerate surfaces in R4, a distinguished transversal bundle called affine normal plane bundle was proposed in [Nomizu-Vrancken]. Lagrangian surfaces have remarkable properties with respect to this normal bundle, like for example, the normal bundle being Lagrangian. In this paper we characterize those surfa…
Laundry surfaces for closed braid diagrams are presented. It is shown that braid diagrams are characterized by linking matrices obtained by lifting cycles from these surfaces. Oriented link types are then characterized by equivalence classes of linking matrices. Similar equivalence classes can be composed of Gordon and…
A major breakthrough in the theory of topological algorithms occurred in 1992 when Hyam Rubinstein introduced the idea of an almost normal surface. We explain how almost normal surfaces emerged naturally from the study of geodesics and minimal surfaces. Patterns of stable and unstable geodesics can be used to character…
We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes bet…
Lie minimal surfaces are characterized by differential equations of principal curvatures.
problem Characterizing Lie minimal surfaces in Riemannian space forms.
method Using Euler-Lagrange equations and differential equations of principal curvatures.
result Rotational surfaces are found for certain relationships between principal curvatures.
Characterizes totally elliptic surface group representations into Lie groups.
problem Understanding totally elliptic surface group representations into Lie groups.
method Characterization of representations into PSL2R and PSL2C by their mapping properties. result They are either into a compact subgroup or Deroin--Tholozan representations.
Characterizes rotational solitons for curve shortening flow on revolution surfaces.
problem Understanding the behavior of curves under curve shortening flow on revolution surfaces.
method Characterization and asymptotic behavior analysis.
result Asymptotic behavior of rotational solitons to parallel geodesics.
Characterizes metrics on triangulated surfaces using glued Euclidean triangles.
problem Describing metrics on triangulated surfaces constructed from glued Euclidean triangles.
method Carefully constructing polyhedral metrics and proving their uniqueness.
result Polyhedral metrics are the only intrinsic metrics preserving Euclidean triangle lengths.
New method characterizes minimal surfaces in 3D space.
problem Characterizing minimal surfaces in 3D space.
method Alexandrov Reflection Method
result Embedded minimal free boundary annuli in B3 are the critical catenoid. Study timelike minimal surfaces in Heisenberg group using harmonic maps.
problem Characterize timelike minimal surfaces in the Heisenberg group.
method Characterization through non-conformal Lorentz harmonic maps and generalized Weierstrass type representation.
result Established generalized Weierstrass type representation for timelike minimal surfaces.
Characterizes curves with short representatives on hyperbolic surfaces.
problem Inequalities on lengths of curves on hyperbolic surfaces.
method Characterization of topological types of curves and multicurves with short representatives.
result Characterizes which topological types of curves and multicurves always have a short representative.
Characterizes transverse surfaces for pseudo-Anosov flows in 3-manifolds.
problem Characterizing surfaces transverse to pseudo-Anosov flows.
method Correspondence between surfaces and veering triangulations, Thurston norm minimization.
result Thurston-norm minimizing surfaces are almost transverse to pseudo-Anosov flows.
Isophote comprises a locus of the surface points whose normal vectors make a constant angle with a fixed vector. Main objective of this paper is to find the axis of an isophote curve via its Darboux frame and afterwards to give some characterizations about isophote and its axis. Particularly, for isophotes lying on a c…
In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.
Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and o…
Sharp characterization of Willmore invariant in higher dimensions.
problem Understanding the Willmore invariant in various dimensions.
method Characterization using conformal fundamental forms and tensors.
result Sharp sufficient condition for vanishing Willmore invariant in even dimensions.
Develops a method to define and characterize geodesics on hyperbolic surfaces.
problem Characterizing closed geodesics on hyperbolic surfaces without self-intersection.
method Constructive definition of the Goldman bracket using closed geodesics.
result Algebraic characterization of geodesics on hyperbolic surfaces.
The study characterizes loxodromes on specific rotational surfaces in 3D space.
problem Characterizing loxodromes on rotational surfaces with special geometric properties.
method Parametrizations and curvature/torsion calculations for loxodromes on various rotational surfaces.
result The loxodrome on a flat rotational surface is a general helix.
Study characterizes bladder motion using dynamic MRI and statistical analysis.
problem Limited volume coverage in dynamic MRI sequences hinders 3D shape reconstruction.
method 3D dense velocity measurements, LDDMM framework, statistical characterization, mean curvature changes, surface deformation analysis.
result Stable shape descriptor for characterizing bladder surface dynamics.